# On existence of directly finite, only one sided self-injective regular rings

- Authors
- Journal
- Journal of Algebra 0021-8693
- Publisher
- Elsevier
- Publication Date
- Volume
- 144
- Issue
- 1
- Identifiers
- DOI: 10.1016/0021-8693(91)90131-q

## Abstract

Abstract This paper is concerned with the following open problem. Is every directly finite, regular right self-injective ring necessarily left self-injective [2, Open problem 14]? This problem is 21 years old and came from J.-E. Roos [8]. We solve in the negative by giving an example. In the example, we construct a simple regular subring of the rank metric completion of a direct limit of a sequence of matrix rings over a field. Moreover the maximal right quotient ring of this simple regular ring is directly finite and not left self-injective. This result was announced in [6] without proof.

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